LMFDB - Embedded newform 120.3.l.a.41.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,3,Mod(41,120)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("120.41");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	120.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.26976317232$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.681615360000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 $ x^8 - 4x^7 - 2x^6 + 20x^5 + 49x^4 - 136x^3 + 168x^2 - 96*x + 864
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.5
Root			$-2.22255 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	120.41
Dual form			120.3.l.a.41.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.291610 - 2.98579i) q^{3} -2.23607i q^{5} +4.46268 q^{7} +(-8.82993 - 1.74137i) q^{9} +O(q^{10})$	\(q+(0.291610 - 2.98579i) q^{3} -2.23607i q^{5} +4.46268 q^{7} +(-8.82993 - 1.74137i) q^{9} -17.8696i q^{11} -11.0107 q^{13} +(-6.67644 - 0.652060i) q^{15} +0.794055i q^{17} +26.5852 q^{19} +(1.30136 - 13.3246i) q^{21} +14.9276i q^{23} -5.00000 q^{25} +(-7.77428 + 25.8565i) q^{27} +5.58545i q^{29} +53.1074 q^{31} +(-53.3549 - 5.21095i) q^{33} -9.97885i q^{35} +51.7565 q^{37} +(-3.21084 + 32.8758i) q^{39} +67.8236i q^{41} -40.8243 q^{43} +(-3.89383 + 19.7443i) q^{45} +12.3483i q^{47} -29.0845 q^{49} +(2.37088 + 0.231554i) q^{51} +37.0351i q^{53} -39.9576 q^{55} +(7.75251 - 79.3779i) q^{57} -61.0932i q^{59} +97.8289 q^{61} +(-39.4051 - 7.77119i) q^{63} +24.6208i q^{65} +3.02541 q^{67} +(44.5708 + 4.35305i) q^{69} -57.0787i q^{71} -31.4690 q^{73} +(-1.45805 + 14.9290i) q^{75} -79.7461i q^{77} -2.16053 q^{79} +(74.9352 + 30.7524i) q^{81} -13.0710i q^{83} +1.77556 q^{85} +(16.6770 + 1.62877i) q^{87} -173.692i q^{89} -49.1374 q^{91} +(15.4866 - 158.568i) q^{93} -59.4463i q^{95} -91.6381 q^{97} +(-31.1176 + 157.787i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9	$ 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} - 8 q^{13} - 8 q^{19} + 28 q^{21} - 40 q^{25} + 20 q^{27} + 120 q^{31} - 112 q^{33} + 8 q^{37} - 72 q^{39} - 328 q^{43} - 60 q^{45} + 64 q^{49} + 64 q^{51} - 40 q^{55} + 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} + 32 q^{73} + 20 q^{75} + 88 q^{79} + 224 q^{81} - 152 q^{87} + 560 q^{91} - 368 q^{93} + 144 q^{97} + 32 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9 - 8 * q^13 - 8 * q^19 + 28 * q^21 - 40 * q^25 + 20 * q^27 + 120 * q^31 - 112 * q^33 + 8 * q^37 - 72 * q^39 - 328 * q^43 - 60 * q^45 + 64 * q^49 + 64 * q^51 - 40 * q^55 + 72 * q^57 + 8 * q^61 + 88 * q^63 + 152 * q^67 + 100 * q^69 + 32 * q^73 + 20 * q^75 + 88 * q^79 + 224 * q^81 - 152 * q^87 + 560 * q^91 - 368 * q^93 + 144 * q^97 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$.

$n$	$31$	$41$	$61$	$97$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.291610	−	2.98579i	0.0972033	−	0.995265i
$3$	0.291610	−	2.98579i	0.0972033	−	0.995265i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	4.46268			0.637525			0.318763	−	0.947835i	$-0.396733\pi$
$7$	4.46268			0.637525			0.318763	+	0.947835i	$0.396733\pi$
$8$		0			0
$9$	−8.82993	−	1.74137i	−0.981103	−	0.193486i
$10$		0			0
$11$		−	17.8696i		−	1.62451i	−0.583305	−	0.812253i	$-0.698241\pi$
$11$		−	17.8696i		−	1.62451i	0.583305	−	0.812253i	$-0.301759\pi$
$12$		0			0
$13$	−11.0107			−0.846980			−0.423490	−	0.905901i	$-0.639195\pi$
$13$	−11.0107			−0.846980			−0.423490	+	0.905901i	$0.639195\pi$
$14$		0			0
$15$	−6.67644	−	0.652060i	−0.445096	−	0.0434707i
$16$		0			0
$17$			0.794055i			0.0467091i	0.999727	+	0.0233546i	$0.00743466\pi$
$17$			0.794055i			0.0467091i	−0.999727	+	0.0233546i	$0.992565\pi$
$18$		0			0
$19$	26.5852			1.39922			0.699611	−	0.714524i	$-0.253357\pi$
$19$	26.5852			1.39922			0.699611	+	0.714524i	$0.253357\pi$
$20$		0			0
$21$	1.30136	−	13.3246i	0.0619696	−	0.634506i
$22$		0			0
$23$			14.9276i			0.649027i	0.945881	+	0.324514i	$0.105201\pi$
$23$			14.9276i			0.649027i	−0.945881	+	0.324514i	$0.894799\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−7.77428	+	25.8565i	−0.287936	+	0.957650i
$28$		0			0
$29$			5.58545i			0.192602i	0.995352	+	0.0963008i	$0.0307011\pi$
$29$			5.58545i			0.192602i	−0.995352	+	0.0963008i	$0.969299\pi$
$30$		0			0
$31$	53.1074			1.71314			0.856571	−	0.516030i	$-0.172591\pi$
$31$	53.1074			1.71314			0.856571	+	0.516030i	$0.172591\pi$
$32$		0			0
$33$	−53.3549	−	5.21095i	−1.61681	−	0.157907i
$34$		0			0
$35$		−	9.97885i		−	0.285110i
$36$		0			0
$37$	51.7565			1.39882			0.699412	−	0.714719i	$-0.253445\pi$
$37$	51.7565			1.39882			0.699412	+	0.714719i	$0.253445\pi$
$38$		0			0
$39$	−3.21084	+	32.8758i	−0.0823293	+	0.842969i
$40$		0			0
$41$			67.8236i			1.65423i	0.562030	+	0.827117i	$0.310020\pi$
$41$			67.8236i			1.65423i	−0.562030	+	0.827117i	$0.689980\pi$
$42$		0			0
$43$	−40.8243			−0.949403			−0.474701	−	0.880147i	$-0.657444\pi$
$43$	−40.8243			−0.949403			−0.474701	+	0.880147i	$0.657444\pi$
$44$		0			0
$45$	−3.89383	+	19.7443i	−0.0865296	+	0.438763i
$46$		0			0
$47$			12.3483i			0.262729i	0.991334	+	0.131365i	$0.0419359\pi$
$47$			12.3483i			0.262729i	−0.991334	+	0.131365i	$0.958064\pi$
$48$		0			0
$49$	−29.0845			−0.593562
$50$		0			0
$51$	2.37088	+	0.231554i	0.0464879	+	0.00454028i
$52$		0			0
$53$			37.0351i			0.698775i	0.936978	+	0.349387i	$0.113610\pi$
$53$			37.0351i			0.698775i	−0.936978	+	0.349387i	$0.886390\pi$
$54$		0			0
$55$	−39.9576			−0.726502
$56$		0			0
$57$	7.75251	−	79.3779i	0.136009	−	1.39260i
$58$		0			0
$59$		−	61.0932i		−	1.03548i	−0.855539	−	0.517739i	$-0.826774\pi$
$59$		−	61.0932i		−	1.03548i	0.855539	−	0.517739i	$-0.173226\pi$
$60$		0			0
$61$	97.8289			1.60375			0.801876	−	0.597490i	$-0.203835\pi$
$61$	97.8289			1.60375			0.801876	+	0.597490i	$0.203835\pi$
$62$		0			0
$63$	−39.4051	−	7.77119i	−0.625478	−	0.123352i
$64$		0			0
$65$			24.6208i			0.378781i
$66$		0			0
$67$	3.02541			0.0451553			0.0225777	−	0.999745i	$-0.492813\pi$
$67$	3.02541			0.0451553			0.0225777	+	0.999745i	$0.492813\pi$
$68$		0			0
$69$	44.5708	+	4.35305i	0.645954	+	0.0630876i
$70$		0			0
$71$		−	57.0787i		−	0.803926i	−0.915656	−	0.401963i	$-0.868328\pi$
$71$		−	57.0787i		−	0.803926i	0.915656	−	0.401963i	$-0.131672\pi$
$72$		0			0
$73$	−31.4690			−0.431082			−0.215541	−	0.976495i	$-0.569152\pi$
$73$	−31.4690			−0.431082			−0.215541	+	0.976495i	$0.569152\pi$
$74$		0			0
$75$	−1.45805	+	14.9290i	−0.0194407	+	0.199053i
$76$		0			0
$77$		−	79.7461i		−	1.03566i
$78$		0			0
$79$	−2.16053			−0.0273485			−0.0136743	−	0.999907i	$-0.504353\pi$
$79$	−2.16053			−0.0273485			−0.0136743	+	0.999907i	$0.504353\pi$
$80$		0			0
$81$	74.9352	+	30.7524i	0.925126	+	0.379660i
$82$		0			0
$83$		−	13.0710i		−	0.157482i	−0.996895	−	0.0787411i	$-0.974910\pi$
$83$		−	13.0710i		−	0.157482i	0.996895	−	0.0787411i	$-0.0250900\pi$
$84$		0			0
$85$	1.77556			0.0208890
$86$		0			0
$87$	16.6770	+	1.62877i	0.191690	+	0.0187215i
$88$		0			0
$89$		−	173.692i		−	1.95160i	−0.218667	−	0.975800i	$-0.570171\pi$
$89$		−	173.692i		−	1.95160i	0.218667	−	0.975800i	$-0.429829\pi$
$90$		0			0
$91$	−49.1374			−0.539971
$92$		0			0
$93$	15.4866	−	158.568i	0.166523	−	1.70503i
$94$		0			0
$95$		−	59.4463i		−	0.625751i
$96$		0			0
$97$	−91.6381			−0.944722			−0.472361	−	0.881405i	$-0.656598\pi$
$97$	−91.6381			−0.944722			−0.472361	+	0.881405i	$0.656598\pi$
$98$		0			0
$99$	−31.1176	+	157.787i	−0.314319	+	1.59381i
$100$		0			0
$101$			116.353i			1.15201i	0.817446	+	0.576005i	$0.195389\pi$
$101$			116.353i			1.15201i	−0.817446	+	0.576005i	$0.804611\pi$
$102$		0			0
$103$	−182.071			−1.76768			−0.883842	−	0.467786i	$-0.845052\pi$
$103$	−182.071			−1.76768			−0.883842	+	0.467786i	$0.845052\pi$
$104$		0			0
$105$	−29.7948	−	2.90993i	−0.283760	−	0.0277136i
$106$		0			0
$107$			102.828i			0.961013i	0.876991	+	0.480506i	$0.159547\pi$
$107$			102.828i			0.961013i	−0.876991	+	0.480506i	$0.840453\pi$
$108$		0			0
$109$	75.4257			0.691979			0.345990	−	0.938238i	$-0.387543\pi$
$109$	75.4257			0.691979			0.345990	+	0.938238i	$0.387543\pi$
$110$		0			0
$111$	15.0927	−	154.534i	0.135970	−	1.39220i
$112$		0			0
$113$			141.923i			1.25596i	0.778230	+	0.627980i	$0.216118\pi$
$113$			141.923i			1.25596i	−0.778230	+	0.627980i	$0.783882\pi$
$114$		0			0
$115$	33.3792			0.290254
$116$		0			0
$117$	97.2241	+	19.1738i	0.830975	+	0.163879i
$118$		0			0
$119$			3.54361i			0.0297782i
$120$		0			0
$121$	−198.322			−1.63902
$122$		0			0
$123$	202.507	+	19.7780i	1.64640	+	0.160797i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	112.920			0.889132			0.444566	−	0.895746i	$-0.353358\pi$
$127$	112.920			0.889132			0.444566	+	0.895746i	$0.353358\pi$
$128$		0			0
$129$	−11.9048	+	121.893i	−0.0922851	+	0.944907i
$130$		0			0
$131$		−	107.233i		−	0.818570i	−0.912407	−	0.409285i	$-0.865778\pi$
$131$		−	107.233i		−	0.818570i	0.912407	−	0.409285i	$-0.134222\pi$
$132$		0			0
$133$	118.641			0.892039
$134$		0			0
$135$	57.8170	+	17.3838i	0.428274	+	0.128769i
$136$		0			0
$137$			67.7434i			0.494477i	0.968955	+	0.247239i	$0.0795232\pi$
$137$			67.7434i			0.494477i	−0.968955	+	0.247239i	$0.920477\pi$
$138$		0			0
$139$	−141.204			−1.01586			−0.507928	−	0.861399i	$-0.669589\pi$
$139$	−141.204			−1.01586			−0.507928	+	0.861399i	$0.669589\pi$
$140$		0			0
$141$	36.8694	+	3.60088i	0.261485	+	0.0255382i
$142$		0			0
$143$			196.757i			1.37593i
$144$		0			0
$145$	12.4894			0.0861341
$146$		0			0
$147$	−8.48134	+	86.8404i	−0.0576962	+	0.590751i
$148$		0			0
$149$			55.0831i			0.369685i	0.982768	+	0.184843i	$0.0591775\pi$
$149$			55.0831i			0.369685i	−0.982768	+	0.184843i	$0.940822\pi$
$150$		0			0
$151$	56.1302			0.371723			0.185862	−	0.982576i	$-0.440492\pi$
$151$	56.1302			0.371723			0.185862	+	0.982576i	$0.440492\pi$
$152$		0			0
$153$	1.38275	−	7.01145i	0.00903756	−	0.0458265i
$154$		0			0
$155$		−	118.752i		−	0.766140i
$156$		0			0
$157$	−274.266			−1.74692			−0.873460	−	0.486896i	$-0.838129\pi$
$157$	−274.266			−1.74692			−0.873460	+	0.486896i	$0.838129\pi$
$158$		0			0
$159$	110.579	+	10.7998i	0.695466	+	0.0679232i
$160$		0			0
$161$			66.6172i			0.413771i
$162$		0			0
$163$	260.316			1.59703			0.798516	−	0.601974i	$-0.205619\pi$
$163$	260.316			1.59703			0.798516	+	0.601974i	$0.205619\pi$
$164$		0			0
$165$	−11.6520	+	119.305i	−0.0706184	+	0.723061i
$166$		0			0
$167$			179.949i			1.07754i	0.842453	+	0.538769i	$0.181111\pi$
$167$			179.949i			1.07754i	−0.842453	+	0.538769i	$0.818889\pi$
$168$		0			0
$169$	−47.7635			−0.282624
$170$		0			0
$171$	−234.745	−	46.2948i	−1.37278	−	0.270730i
$172$		0			0
$173$		−	111.265i		−	0.643148i	−0.946884	−	0.321574i	$-0.895788\pi$
$173$		−	111.265i		−	0.643148i	0.946884	−	0.321574i	$-0.104212\pi$
$174$		0			0
$175$	−22.3134			−0.127505
$176$		0			0
$177$	−182.412	−	17.8154i	−1.03057	−	0.100652i
$178$		0			0
$179$		−	77.8475i		−	0.434902i	−0.976071	−	0.217451i	$-0.930226\pi$
$179$		−	77.8475i		−	0.434902i	0.976071	−	0.217451i	$-0.0697743\pi$
$180$		0			0
$181$	−238.852			−1.31963			−0.659813	−	0.751430i	$-0.729364\pi$
$181$	−238.852			−1.31963			−0.659813	+	0.751430i	$0.729364\pi$
$182$		0			0
$183$	28.5279	−	292.097i	0.155890	−	1.59616i
$184$		0			0
$185$		−	115.731i		−	0.625573i
$186$		0			0
$187$	14.1894			0.0758793
$188$		0			0
$189$	−34.6941	+	115.389i	−0.183567	+	0.610526i
$190$		0			0
$191$			177.248i			0.928002i	0.885835	+	0.464001i	$0.153587\pi$
$191$			177.248i			0.928002i	−0.885835	+	0.464001i	$0.846413\pi$
$192$		0			0
$193$	284.254			1.47282			0.736409	−	0.676536i	$-0.236520\pi$
$193$	284.254			1.47282			0.736409	+	0.676536i	$0.236520\pi$
$194$		0			0
$195$	73.5125	+	7.17966i	0.376987	+	0.0368188i
$196$		0			0
$197$		−	244.341i		−	1.24031i	−0.784479	−	0.620156i	$-0.787069\pi$
$197$		−	244.341i		−	1.24031i	0.784479	−	0.620156i	$-0.212931\pi$
$198$		0			0
$199$	74.0122			0.371921			0.185960	−	0.982557i	$-0.440460\pi$
$199$	74.0122			0.371921			0.185960	+	0.982557i	$0.440460\pi$
$200$		0			0
$201$	0.882239	−	9.03324i	0.00438925	−	0.0449415i
$202$		0			0
$203$			24.9260i			0.122788i
$204$		0			0
$205$	151.658			0.739796
$206$		0			0
$207$	25.9946	−	131.810i	0.125578	−	0.636763i
$208$		0			0
$209$		−	475.066i		−	2.27304i
$210$		0			0
$211$	−31.0682			−0.147243			−0.0736214	−	0.997286i	$-0.523456\pi$
$211$	−31.0682			−0.147243			−0.0736214	+	0.997286i	$0.523456\pi$
$212$		0			0
$213$	−170.425	−	16.6447i	−0.800119	−	0.0781443i
$214$		0			0
$215$			91.2859i			0.424586i
$216$		0			0
$217$	237.001			1.09217
$218$		0			0
$219$	−9.17668	+	93.9600i	−0.0419026	+	0.429041i
$220$		0			0
$221$		−	8.74314i		−	0.0395617i
$222$		0			0
$223$	100.432			0.450366			0.225183	−	0.974316i	$-0.427702\pi$
$223$	100.432			0.450366			0.225183	+	0.974316i	$0.427702\pi$
$224$		0			0
$225$	44.1496	+	8.70687i	0.196221	+	0.0386972i
$226$		0			0
$227$		−	349.117i		−	1.53796i	−0.639273	−	0.768980i	$-0.720764\pi$
$227$		−	349.117i		−	1.53796i	0.639273	−	0.768980i	$-0.279236\pi$
$228$		0			0
$229$	−165.007			−0.720555			−0.360278	−	0.932845i	$-0.617318\pi$
$229$	−165.007			−0.720555			−0.360278	+	0.932845i	$0.617318\pi$
$230$		0			0
$231$	−238.105	−	23.2548i	−1.03076	−	0.100670i
$232$		0			0
$233$			124.800i			0.535620i	0.963472	+	0.267810i	$0.0863000\pi$
$233$			124.800i			0.535620i	−0.963472	+	0.267810i	$0.913700\pi$
$234$		0			0
$235$	27.6116			0.117496
$236$		0			0
$237$	−0.630033	+	6.45091i	−0.00265837	+	0.0272190i
$238$		0			0
$239$		−	61.8321i		−	0.258712i	−0.991598	−	0.129356i	$-0.958709\pi$
$239$		−	61.8321i		−	0.258712i	0.991598	−	0.129356i	$-0.0412910\pi$
$240$		0			0
$241$	2.25555			0.00935913			0.00467956	−	0.999989i	$-0.498510\pi$
$241$	2.25555			0.00935913			0.00467956	+	0.999989i	$0.498510\pi$
$242$		0			0
$243$	113.672	−	214.773i	0.467787	−	0.883841i
$244$		0			0
$245$			65.0350i			0.265449i
$246$		0			0
$247$	−292.723			−1.18511
$248$		0			0
$249$	−39.0274	−	3.81164i	−0.156736	−	0.0153078i
$250$		0			0
$251$			108.993i			0.434234i	0.976146	+	0.217117i	$0.0696654\pi$
$251$			108.993i			0.434234i	−0.976146	+	0.217117i	$0.930335\pi$
$252$		0			0
$253$	266.750			1.05435
$254$		0			0
$255$	0.517771	−	5.30146i	0.00203048	−	0.0207900i
$256$		0			0
$257$			31.4664i			0.122437i	0.998124	+	0.0612187i	$0.0194987\pi$
$257$			31.4664i			0.122437i	−0.998124	+	0.0612187i	$0.980501\pi$
$258$		0			0
$259$	230.972			0.891785
$260$		0			0
$261$	9.72636	−	49.3191i	0.0372657	−	0.188962i
$262$		0			0
$263$			235.190i			0.894260i	0.894469	+	0.447130i	$0.147554\pi$
$263$			235.190i			0.894260i	−0.894469	+	0.447130i	$0.852446\pi$
$264$		0			0
$265$	82.8129			0.312502
$266$		0			0
$267$	−518.609	−	50.6504i	−1.94236	−	0.189702i
$268$		0			0
$269$			221.066i			0.821808i	0.911679	+	0.410904i	$0.134787\pi$
$269$			221.066i			0.821808i	−0.911679	+	0.410904i	$0.865213\pi$
$270$		0			0
$271$	268.830			0.991991			0.495995	−	0.868325i	$-0.334803\pi$
$271$	268.830			0.991991			0.495995	+	0.868325i	$0.334803\pi$
$272$		0			0
$273$	−14.3290	+	146.714i	−0.0524870	+	0.537414i
$274$		0			0
$275$			89.3479i			0.324901i
$276$		0			0
$277$	144.697			0.522370			0.261185	−	0.965289i	$-0.415887\pi$
$277$	144.697			0.522370			0.261185	+	0.965289i	$0.415887\pi$
$278$		0			0
$279$	−468.934	−	92.4798i	−1.68077	−	0.331469i
$280$		0			0
$281$			189.426i			0.674112i	0.941485	+	0.337056i	$0.109431\pi$
$281$			189.426i			0.674112i	−0.941485	+	0.337056i	$0.890569\pi$
$282$		0			0
$283$	−420.621			−1.48629			−0.743146	−	0.669129i	$-0.766667\pi$
$283$	−420.621			−1.48629			−0.743146	+	0.669129i	$0.766667\pi$
$284$		0			0
$285$	−177.494	−	17.3351i	−0.622788	−	0.0608251i
$286$		0			0
$287$			302.675i			1.05462i
$288$		0			0
$289$	288.369			0.997818
$290$		0			0
$291$	−26.7226	+	273.612i	−0.0918302	+	0.940249i
$292$		0			0
$293$		−	283.828i		−	0.968696i	−0.874875	−	0.484348i	$-0.839057\pi$
$293$		−	283.828i		−	0.968696i	0.874875	−	0.484348i	$-0.160943\pi$
$294$		0			0
$295$	−136.609			−0.463080
$296$		0			0
$297$	462.045	+	138.923i	1.55571	+	0.467754i
$298$		0			0
$299$		−	164.364i		−	0.549713i
$300$		0			0
$301$	−182.186			−0.605268
$302$		0			0
$303$	347.406	+	33.9297i	1.14656	+	0.111979i
$304$		0			0
$305$		−	218.752i		−	0.717220i
$306$		0			0
$307$	167.979			0.547162			0.273581	−	0.961849i	$-0.411792\pi$
$307$	167.979			0.547162			0.273581	+	0.961849i	$0.411792\pi$
$308$		0			0
$309$	−53.0938	+	543.628i	−0.171825	+	1.75931i
$310$		0			0
$311$			192.970i			0.620482i	0.950658	+	0.310241i	$0.100410\pi$
$311$			192.970i			0.620482i	−0.950658	+	0.310241i	$0.899590\pi$
$312$		0			0
$313$	161.056			0.514555			0.257278	−	0.966338i	$-0.417175\pi$
$313$	161.056			0.514555			0.257278	+	0.966338i	$0.417175\pi$
$314$		0			0
$315$	−17.3769	+	88.1125i	−0.0551648	+	0.279722i
$316$		0			0
$317$			514.761i			1.62385i	0.583761	+	0.811926i	$0.301581\pi$
$317$			514.761i			1.62385i	−0.583761	+	0.811926i	$0.698419\pi$
$318$		0			0
$319$	99.8096			0.312883
$320$		0			0
$321$	307.024	+	29.9858i	0.956462	+	0.0934136i
$322$		0			0
$323$			21.1101i			0.0653564i
$324$		0			0
$325$	55.0537			0.169396
$326$		0			0
$327$	21.9949	−	225.206i	0.0672627	−	0.688702i
$328$		0			0
$329$			55.1064i			0.167497i
$330$		0			0
$331$	63.7466			0.192588			0.0962939	−	0.995353i	$-0.469301\pi$
$331$	63.7466			0.192588			0.0962939	+	0.995353i	$0.469301\pi$
$332$		0			0
$333$	−457.006	−	90.1274i	−1.37239	−	0.270653i
$334$		0			0
$335$		−	6.76502i		−	0.0201941i
$336$		0			0
$337$	−64.2444			−0.190636			−0.0953181	−	0.995447i	$-0.530387\pi$
$337$	−64.2444			−0.190636			−0.0953181	+	0.995447i	$0.530387\pi$
$338$		0			0
$339$	423.754	+	41.3863i	1.25001	+	0.122083i
$340$		0			0
$341$		−	949.006i		−	2.78301i
$342$		0			0
$343$	−348.466			−1.01594
$344$		0			0
$345$	9.73371	−	99.6634i	0.0282136	−	0.288879i
$346$		0			0
$347$			66.1416i			0.190610i	0.995448	+	0.0953049i	$0.0303826\pi$
$347$			66.1416i			0.190610i	−0.995448	+	0.0953049i	$0.969617\pi$
$348$		0			0
$349$	−188.752			−0.540837			−0.270419	−	0.962743i	$-0.587162\pi$
$349$	−188.752			−0.540837			−0.270419	+	0.962743i	$0.587162\pi$
$350$		0			0
$351$	85.6006	−	284.700i	0.243876	−	0.811110i
$352$		0			0
$353$			435.927i			1.23492i	0.786602	+	0.617461i	$0.211839\pi$
$353$			435.927i			1.23492i	−0.786602	+	0.617461i	$0.788161\pi$
$354$		0			0
$355$	−127.632			−0.359527
$356$		0			0
$357$	10.5805	+	1.03335i	0.0296372	+	0.00289454i
$358$		0			0
$359$			188.255i			0.524386i	0.965015	+	0.262193i	$0.0844457\pi$
$359$			188.255i			0.524386i	−0.965015	+	0.262193i	$0.915554\pi$
$360$		0			0
$361$	345.773			0.957820
$362$		0			0
$363$	−57.8326	+	592.148i	−0.159318	+	1.63126i
$364$		0			0
$365$			70.3668i			0.192786i
$366$		0			0
$367$	276.868			0.754409			0.377205	−	0.926130i	$-0.376885\pi$
$367$	276.868			0.754409			0.377205	+	0.926130i	$0.376885\pi$
$368$		0			0
$369$	118.106	−	598.877i	0.320071	−	1.62297i
$370$		0			0
$371$			165.275i			0.445486i
$372$		0			0
$373$	−510.098			−1.36756			−0.683778	−	0.729691i	$-0.739664\pi$
$373$	−510.098			−1.36756			−0.683778	+	0.729691i	$0.739664\pi$
$374$		0			0
$375$	33.3822	+	3.26030i	0.0890192	+	0.00869413i
$376$		0			0
$377$		−	61.4999i		−	0.163130i
$378$		0			0
$379$	−298.461			−0.787497			−0.393748	−	0.919218i	$-0.628822\pi$
$379$	−298.461			−0.787497			−0.393748	+	0.919218i	$0.628822\pi$
$380$		0			0
$381$	32.9285	−	337.155i	0.0864266	−	0.884922i
$382$		0			0
$383$		−	500.548i		−	1.30691i	−0.756963	−	0.653457i	$-0.773318\pi$
$383$		−	500.548i		−	1.30691i	0.756963	−	0.653457i	$-0.226682\pi$
$384$		0			0
$385$	−178.318			−0.463163
$386$		0			0
$387$	360.476	+	71.0904i	0.931462	+	0.183696i
$388$		0			0
$389$			465.726i			1.19724i	0.801033	+	0.598620i	$0.204284\pi$
$389$			465.726i			1.19724i	−0.801033	+	0.598620i	$0.795716\pi$
$390$		0			0
$391$	−11.8534			−0.0303155
$392$		0			0
$393$	−320.175	−	31.2701i	−0.814694	−	0.0795678i
$394$		0			0
$395$			4.83110i			0.0122306i
$396$		0			0
$397$	−344.998			−0.869013			−0.434507	−	0.900669i	$-0.643077\pi$
$397$	−344.998			−0.869013			−0.434507	+	0.900669i	$0.643077\pi$
$398$		0			0
$399$	34.5969	−	354.238i	0.0867091	−	0.887815i
$400$		0			0
$401$		−	428.755i		−	1.06922i	−0.845100	−	0.534608i	$-0.820459\pi$
$401$		−	428.755i		−	1.06922i	0.845100	−	0.534608i	$-0.179541\pi$
$402$		0			0
$403$	−584.752			−1.45100
$404$		0			0
$405$	68.7645	−	167.560i	0.169789	−	0.413729i
$406$		0			0
$407$		−	924.866i		−	2.27240i
$408$		0			0
$409$	−349.445			−0.854389			−0.427194	−	0.904160i	$-0.640498\pi$
$409$	−349.445			−0.854389			−0.427194	+	0.904160i	$0.640498\pi$
$410$		0			0
$411$	202.268	+	19.7547i	0.492136	+	0.0480648i
$412$		0			0
$413$		−	272.639i		−	0.660143i
$414$		0			0
$415$	−29.2277			−0.0704281
$416$		0			0
$417$	−41.1765	+	421.606i	−0.0987447	+	1.01105i
$418$		0			0
$419$		−	225.745i		−	0.538770i	−0.963033	−	0.269385i	$-0.913180\pi$
$419$		−	225.745i		−	0.538770i	0.963033	−	0.269385i	$-0.0868205\pi$
$420$		0			0
$421$	−138.931			−0.330003			−0.165002	−	0.986293i	$-0.552763\pi$
$421$	−138.931			−0.330003			−0.165002	+	0.986293i	$0.552763\pi$
$422$		0			0
$423$	21.5030	−	109.034i	0.0508345	−	0.257765i
$424$		0			0
$425$		−	3.97027i		−	0.00934182i
$426$		0			0
$427$	436.579			1.02243
$428$		0			0
$429$	587.477	+	57.3764i	1.36941	+	0.133745i
$430$		0			0
$431$			289.802i			0.672394i	0.941792	+	0.336197i	$0.109141\pi$
$431$			289.802i			0.672394i	−0.941792	+	0.336197i	$0.890859\pi$
$432$		0			0
$433$	53.8425			0.124348			0.0621738	−	0.998065i	$-0.480197\pi$
$433$	53.8425			0.124348			0.0621738	+	0.998065i	$0.480197\pi$
$434$		0			0
$435$	3.64205	−	37.2909i	0.00837252	−	0.0857262i
$436$		0			0
$437$			396.854i			0.908133i
$438$		0			0
$439$	333.179			0.758949			0.379474	−	0.925202i	$-0.376105\pi$
$439$	333.179			0.758949			0.379474	+	0.925202i	$0.376105\pi$
$440$		0			0
$441$	256.814	+	50.6471i	0.582345	+	0.114846i
$442$		0			0
$443$		−	652.015i		−	1.47182i	−0.677081	−	0.735909i	$-0.736755\pi$
$443$		−	652.015i		−	1.47182i	0.677081	−	0.735909i	$-0.263245\pi$
$444$		0			0
$445$	−388.388			−0.872782
$446$		0			0
$447$	164.467	+	16.0628i	0.367935	+	0.0359346i
$448$		0			0
$449$		−	115.054i		−	0.256245i	−0.991758	−	0.128122i	$-0.959105\pi$
$449$		−	115.054i		−	0.256245i	0.991758	−	0.128122i	$-0.0408950\pi$
$450$		0			0
$451$	1211.98			2.68731
$452$		0			0
$453$	16.3681	−	167.593i	0.0361327	−	0.369963i
$454$		0			0
$455$			109.875i			0.241482i
$456$		0			0
$457$	−138.318			−0.302666			−0.151333	−	0.988483i	$-0.548357\pi$
$457$	−138.318			−0.302666			−0.151333	+	0.988483i	$0.548357\pi$
$458$		0			0
$459$	−20.5315	−	6.17321i	−0.0447310	−	0.0134493i
$460$		0			0
$461$			300.475i			0.651789i	0.945406	+	0.325894i	$0.105665\pi$
$461$			300.475i			0.651789i	−0.945406	+	0.325894i	$0.894335\pi$
$462$		0			0
$463$	−91.5721			−0.197780			−0.0988899	−	0.995098i	$-0.531529\pi$
$463$	−91.5721			−0.197780			−0.0988899	+	0.995098i	$0.531529\pi$
$464$		0			0
$465$	−354.568	−	34.6292i	−0.762512	−	0.0744714i
$466$		0			0
$467$		−	308.136i		−	0.659821i	−0.944012	−	0.329910i	$-0.892981\pi$
$467$		−	308.136i		−	0.659821i	0.944012	−	0.329910i	$-0.107019\pi$
$468$		0			0
$469$	13.5014			0.0287877
$470$		0			0
$471$	−79.9788	+	818.903i	−0.169806	+	1.73865i
$472$		0			0
$473$			729.513i			1.54231i
$474$		0			0
$475$	−132.926			−0.279844
$476$		0			0
$477$	64.4919	−	327.017i	0.135203	−	0.685570i
$478$		0			0
$479$			218.332i			0.455808i	0.973684	+	0.227904i	$0.0731873\pi$
$479$			218.332i			0.455808i	−0.973684	+	0.227904i	$0.926813\pi$
$480$		0			0
$481$	−569.877			−1.18478
$482$		0			0
$483$	198.905	+	19.4262i	0.411812	+	0.0402200i
$484$		0			0
$485$			204.909i			0.422493i
$486$		0			0
$487$	−538.368			−1.10548			−0.552739	−	0.833354i	$-0.686417\pi$
$487$	−538.368			−1.10548			−0.552739	+	0.833354i	$0.686417\pi$
$488$		0			0
$489$	75.9108	−	777.250i	0.155237	−	1.58947i
$490$		0			0
$491$		−	418.223i		−	0.851777i	−0.904776	−	0.425889i	$-0.859962\pi$
$491$		−	418.223i		−	0.851777i	0.904776	−	0.425889i	$-0.140038\pi$
$492$		0			0
$493$	−4.43515			−0.00899625
$494$		0			0
$495$	352.823	+	69.5811i	0.712773	+	0.140568i
$496$		0			0
$497$		−	254.724i		−	0.512523i
$498$		0			0
$499$	−732.199			−1.46733			−0.733667	−	0.679510i	$-0.762193\pi$
$499$	−732.199			−1.46733			−0.733667	+	0.679510i	$0.762193\pi$
$500$		0			0
$501$	537.291	+	52.4749i	1.07244	+	0.104740i
$502$		0			0
$503$			338.339i			0.672641i	0.941748	+	0.336321i	$0.109183\pi$
$503$			338.339i			0.672641i	−0.941748	+	0.336321i	$0.890817\pi$
$504$		0			0
$505$	260.173			0.515195
$506$		0			0
$507$	−13.9283	+	142.612i	−0.0274720	+	0.281286i
$508$		0			0
$509$			864.760i			1.69894i	0.527637	+	0.849470i	$0.323078\pi$
$509$			864.760i			1.69894i	−0.527637	+	0.849470i	$0.676922\pi$
$510$		0			0
$511$	−140.436			−0.274826
$512$		0			0
$513$	−206.681	+	687.401i	−0.402887	+	1.33996i
$514$		0			0
$515$			407.124i			0.790532i
$516$		0			0
$517$	220.659			0.426806
$518$		0			0
$519$	−332.213	−	32.4459i	−0.640103	−	0.0625162i
$520$		0			0
$521$			294.647i			0.565541i	0.959188	+	0.282770i	$0.0912534\pi$
$521$			294.647i			0.565541i	−0.959188	+	0.282770i	$0.908747\pi$
$522$		0			0
$523$	−613.850			−1.17371			−0.586855	−	0.809692i	$-0.699634\pi$
$523$	−613.850			−1.17371			−0.586855	+	0.809692i	$0.699634\pi$
$524$		0			0
$525$	−6.50680	+	66.6231i	−0.0123939	+	0.126901i
$526$		0			0
$527$			42.1702i			0.0800193i
$528$		0			0
$529$	306.166			0.578763
$530$		0			0
$531$	−106.386	+	539.448i	−0.200351	+	1.01591i
$532$		0			0
$533$		−	746.788i		−	1.40110i
$534$		0			0
$535$	229.931			0.429778
$536$		0			0
$537$	−232.437	−	22.7011i	−0.432843	−	0.0422740i
$538$		0			0
$539$			519.728i			0.964245i
$540$		0			0
$541$	188.436			0.348311			0.174156	−	0.984718i	$-0.444280\pi$
$541$	188.436			0.348311			0.174156	+	0.984718i	$0.444280\pi$
$542$		0			0
$543$	−69.6517	+	713.163i	−0.128272	+	1.31338i
$544$		0			0
$545$		−	168.657i		−	0.309462i
$546$		0			0
$547$	637.744			1.16589			0.582947	−	0.812510i	$-0.301899\pi$
$547$	637.744			1.16589			0.582947	+	0.812510i	$0.301899\pi$
$548$		0			0
$549$	−863.822	−	170.357i	−1.57345	−	0.310304i
$550$		0			0
$551$			148.490i			0.269492i
$552$		0			0
$553$	−9.64176			−0.0174354
$554$		0			0
$555$	−345.549	−	33.7483i	−0.622611	−	0.0608078i
$556$		0			0
$557$			77.2307i			0.138655i	0.997594	+	0.0693274i	$0.0220853\pi$
$557$			77.2307i			0.138655i	−0.997594	+	0.0693274i	$0.977915\pi$
$558$		0			0
$559$	449.506			0.804125
$560$		0			0
$561$	4.13778	−	42.3667i	0.00737572	−	0.0755200i
$562$		0			0
$563$			257.584i			0.457521i	0.973483	+	0.228760i	$0.0734672\pi$
$563$			257.584i			0.457521i	−0.973483	+	0.228760i	$0.926533\pi$
$564$		0			0
$565$	317.351			0.561682
$566$		0			0
$567$	334.412	+	137.238i	0.589791	+	0.242042i
$568$		0			0
$569$			307.483i			0.540392i	0.962805	+	0.270196i	$0.0870886\pi$
$569$			307.483i			0.540392i	−0.962805	+	0.270196i	$0.912911\pi$
$570$		0			0
$571$	−384.100			−0.672679			−0.336340	−	0.941741i	$-0.609189\pi$
$571$	−384.100			−0.672679			−0.336340	+	0.941741i	$0.609189\pi$
$572$		0			0
$573$	529.227	+	51.6874i	0.923608	+	0.0902049i
$574$		0			0
$575$		−	74.6382i		−	0.129805i
$576$		0			0
$577$	−235.337			−0.407863			−0.203932	−	0.978985i	$-0.565372\pi$
$577$	−235.337			−0.407863			−0.203932	+	0.978985i	$0.565372\pi$
$578$		0			0
$579$	82.8913	−	848.724i	0.143163	−	1.46584i
$580$		0			0
$581$		−	58.3317i		−	0.100399i
$582$		0			0
$583$	661.801			1.13516
$584$		0			0
$585$	42.8740	−	217.400i	0.0732889	−	0.371623i
$586$		0			0
$587$		−	230.275i		−	0.392292i	−0.980575	−	0.196146i	$-0.937157\pi$
$587$		−	230.275i		−	0.392292i	0.980575	−	0.196146i	$-0.0628426\pi$
$588$		0			0
$589$	1411.87			2.39706
$590$		0			0
$591$	−729.553	−	71.2524i	−1.23444	−	0.120562i
$592$		0			0
$593$		−	1042.09i		−	1.75731i	−0.477455	−	0.878656i	$-0.658441\pi$
$593$		−	1042.09i		−	1.75731i	0.477455	−	0.878656i	$-0.341559\pi$
$594$		0			0
$595$	7.92375			0.0133172
$596$		0			0
$597$	21.5827	−	220.985i	0.0361519	−	0.370160i
$598$		0			0
$599$		−	136.181i		−	0.227348i	−0.993518	−	0.113674i	$-0.963738\pi$
$599$		−	136.181i		−	0.227348i	0.993518	−	0.113674i	$-0.0362619\pi$
$600$		0			0
$601$	136.625			0.227329			0.113665	−	0.993519i	$-0.463741\pi$
$601$	136.625			0.227329			0.113665	+	0.993519i	$0.463741\pi$
$602$		0			0
$603$	−26.7141	−	5.26837i	−0.0443020	−	0.00873693i
$604$		0			0
$605$			443.461i			0.732993i
$606$		0			0
$607$	−426.666			−0.702910			−0.351455	−	0.936205i	$-0.614313\pi$
$607$	−426.666			−0.702910			−0.351455	+	0.936205i	$0.614313\pi$
$608$		0			0
$609$	74.4240	+	7.26868i	0.122207	+	0.0119354i
$610$		0			0
$611$		−	135.964i		−	0.222527i
$612$		0			0
$613$	665.317			1.08535			0.542673	−	0.839944i	$-0.317412\pi$
$613$	665.317			1.08535			0.542673	+	0.839944i	$0.317412\pi$
$614$		0			0
$615$	44.2250	−	452.820i	0.0719106	−	0.736292i
$616$		0			0
$617$			826.628i			1.33975i	0.742473	+	0.669877i	$0.233653\pi$
$617$			826.628i			1.33975i	−0.742473	+	0.669877i	$0.766347\pi$
$618$		0			0
$619$	19.8176			0.0320155			0.0160078	−	0.999872i	$-0.494904\pi$
$619$	19.8176			0.0320155			0.0160078	+	0.999872i	$0.494904\pi$
$620$		0			0
$621$	−385.977	−	116.052i	−0.621541	−	0.186879i
$622$		0			0
$623$		−	775.133i		−	1.24419i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−1418.45	−	138.534i	−2.26228	−	0.220948i
$628$		0			0
$629$			41.0975i			0.0653378i
$630$		0			0
$631$	−728.156			−1.15397			−0.576986	−	0.816754i	$-0.695771\pi$
$631$	−728.156			−1.15397			−0.576986	+	0.816754i	$0.695771\pi$
$632$		0			0
$633$	−9.05981	+	92.7633i	−0.0143125	+	0.146546i
$634$		0			0
$635$		−	252.496i		−	0.397632i
$636$		0			0
$637$	320.242			0.502735
$638$		0			0
$639$	−99.3955	+	504.001i	−0.155548	+	0.788734i
$640$		0			0
$641$			438.969i			0.684819i	0.939551	+	0.342410i	$0.111243\pi$
$641$			438.969i			0.684819i	−0.939551	+	0.342410i	$0.888757\pi$
$642$		0			0
$643$	483.535			0.751998			0.375999	−	0.926620i	$-0.377300\pi$
$643$	483.535			0.751998			0.375999	+	0.926620i	$0.377300\pi$
$644$		0			0
$645$	272.561	+	26.6199i	0.422575	+	0.0412711i
$646$		0			0
$647$			575.687i			0.889780i	0.895585	+	0.444890i	$0.146757\pi$
$647$			575.687i			0.889780i	−0.895585	+	0.444890i	$0.853243\pi$
$648$		0			0
$649$	−1091.71			−1.68214
$650$		0			0
$651$	69.1119	−	707.636i	0.106163	−	1.08700i
$652$		0			0
$653$		−	407.885i		−	0.624633i	−0.949978	−	0.312317i	$-0.898895\pi$
$653$		−	407.885i		−	0.624633i	0.949978	−	0.312317i	$-0.101105\pi$
$654$		0			0
$655$	−239.780			−0.366076
$656$		0			0
$657$	277.869	+	54.7993i	0.422936	+	0.0834084i
$658$		0			0
$659$		−	187.060i		−	0.283854i	−0.989877	−	0.141927i	$-0.954670\pi$
$659$		−	187.060i		−	0.283854i	0.989877	−	0.141927i	$-0.0453298\pi$
$660$		0			0
$661$	483.965			0.732171			0.366085	−	0.930581i	$-0.380698\pi$
$661$	483.965			0.732171			0.366085	+	0.930581i	$0.380698\pi$
$662$		0			0
$663$	−26.1052	−	2.54959i	−0.0393744	−	0.00384553i
$664$		0			0
$665$		−	265.290i		−	0.398932i
$666$		0			0
$667$	−83.3775			−0.125004
$668$		0			0
$669$	29.2869	−	299.868i	0.0437771	−	0.448233i
$670$		0			0
$671$		−	1748.16i		−	2.60531i
$672$		0			0
$673$	480.449			0.713891			0.356945	−	0.934125i	$-0.383818\pi$
$673$	480.449			0.713891			0.356945	+	0.934125i	$0.383818\pi$
$674$		0			0
$675$	38.8714	−	129.283i	0.0575873	−	0.191530i
$676$		0			0
$677$		−	1245.12i		−	1.83918i	−0.392881	−	0.919589i	$-0.628522\pi$
$677$		−	1245.12i		−	1.83918i	0.392881	−	0.919589i	$-0.371478\pi$
$678$		0			0
$679$	−408.951			−0.602284
$680$		0			0
$681$	−1042.39	−	101.806i	−1.53068	−	0.149495i
$682$		0			0
$683$		−	118.409i		−	0.173366i	−0.996236	−	0.0866832i	$-0.972373\pi$
$683$		−	118.409i		−	0.173366i	0.996236	−	0.0866832i	$-0.0276268\pi$
$684$		0			0
$685$	151.479			0.221137
$686$		0			0
$687$	−48.1177	+	492.677i	−0.0700404	+	0.717143i
$688$		0			0
$689$		−	407.784i		−	0.591848i
$690$		0			0
$691$	−481.257			−0.696465			−0.348232	−	0.937408i	$-0.613218\pi$
$691$	−481.257			−0.696465			−0.348232	+	0.937408i	$0.613218\pi$
$692$		0			0
$693$	−138.868	+	704.152i	−0.200387	+	1.01609i
$694$		0			0
$695$			315.742i			0.454305i
$696$		0			0
$697$	−53.8556			−0.0772678
$698$		0			0
$699$	372.626	+	36.3928i	0.533084	+	0.0520641i
$700$		0			0
$701$		−	644.879i		−	0.919941i	−0.887934	−	0.459971i	$-0.847860\pi$
$701$		−	644.879i		−	0.919941i	0.887934	−	0.459971i	$-0.152140\pi$
$702$		0			0
$703$	1375.96			1.95726
$704$		0			0
$705$	8.05182	−	82.4425i	0.0114210	−	0.116940i
$706$		0			0
$707$			519.246i			0.734436i
$708$		0			0
$709$	−534.801			−0.754303			−0.377151	−	0.926152i	$-0.623096\pi$
$709$	−534.801			−0.754303			−0.377151	+	0.926152i	$0.623096\pi$
$710$		0			0
$711$	19.0774	+	3.76230i	0.0268317	+	0.00529156i
$712$		0			0
$713$			792.767i			1.11188i
$714$		0			0
$715$	439.963			0.615332
$716$		0			0
$717$	−184.618	−	18.0309i	−0.257487	−	0.0251476i
$718$		0			0
$719$			891.507i			1.23993i	0.784631	+	0.619963i	$0.212852\pi$
$719$			891.507i			1.23993i	−0.784631	+	0.619963i	$0.787148\pi$
$720$		0			0
$721$	−812.526			−1.12694
$722$		0			0
$723$	0.657741	−	6.73461i	0.000909739	−	0.00931481i
$724$		0			0
$725$		−	27.9272i		−	0.0385203i
$726$		0			0
$727$	−720.024			−0.990405			−0.495202	−	0.868778i	$-0.664906\pi$
$727$	−720.024			−0.990405			−0.495202	+	0.868778i	$0.664906\pi$
$728$		0			0
$729$	−608.121	−	402.032i	−0.834185	−	0.551484i
$730$		0			0
$731$		−	32.4167i		−	0.0443458i
$732$		0			0
$733$	953.335			1.30059			0.650296	−	0.759681i	$-0.274645\pi$
$733$	953.335			1.30059			0.650296	+	0.759681i	$0.274645\pi$
$734$		0			0
$735$	194.181	+	18.9649i	0.264192	+	0.0258025i
$736$		0			0
$737$		−	54.0627i		−	0.0733552i
$738$		0			0
$739$	615.723			0.833185			0.416592	−	0.909093i	$-0.363224\pi$
$739$	615.723			0.833185			0.416592	+	0.909093i	$0.363224\pi$
$740$		0			0
$741$	−85.3609	+	874.010i	−0.115197	+	1.17950i
$742$		0			0
$743$		−	377.743i		−	0.508403i	−0.967151	−	0.254201i	$-0.918187\pi$
$743$		−	377.743i		−	0.508403i	0.967151	−	0.254201i	$-0.0818126\pi$
$744$		0			0
$745$	123.170			0.165328
$746$		0			0
$747$	−22.7615	+	115.416i	−0.0304706	+	0.154506i
$748$		0			0
$749$			458.890i			0.612670i
$750$		0			0
$751$	735.578			0.979465			0.489733	−	0.871873i	$-0.337094\pi$
$751$	735.578			0.979465			0.489733	+	0.871873i	$0.337094\pi$
$752$		0			0
$753$	325.430	+	31.7834i	0.432178	+	0.0422090i
$754$		0			0
$755$		−	125.511i		−	0.166240i
$756$		0			0
$757$	921.601			1.21744			0.608719	−	0.793386i	$-0.291684\pi$
$757$	921.601			1.21744			0.608719	+	0.793386i	$0.291684\pi$
$758$		0			0
$759$	77.7871	−	796.462i	0.102486	−	1.04936i
$760$		0			0
$761$		−	240.158i		−	0.315582i	−0.987473	−	0.157791i	$-0.949563\pi$
$761$		−	240.158i		−	0.315582i	0.987473	−	0.157791i	$-0.0504372\pi$
$762$		0			0
$763$	336.601			0.441154
$764$		0			0
$765$	−15.6781	−	3.09192i	−0.0204942	−	0.00404172i
$766$		0			0
$767$			672.682i			0.877029i
$768$		0			0
$769$	−4.79672			−0.00623760			−0.00311880	−	0.999995i	$-0.500993\pi$
$769$	−4.79672			−0.00623760			−0.00311880	+	0.999995i	$0.500993\pi$
$770$		0			0
$771$	93.9522	+	9.17592i	0.121858	+	0.0119013i
$772$		0			0
$773$		−	140.044i		−	0.181170i	−0.995889	−	0.0905850i	$-0.971126\pi$
$773$		−	140.044i		−	0.181170i	0.995889	−	0.0905850i	$-0.0288737\pi$
$774$		0			0
$775$	−265.537			−0.342628
$776$		0			0
$777$	67.3539	−	689.636i	0.0866845	−	0.887562i
$778$		0			0
$779$			1803.10i			2.31464i
$780$		0			0
$781$	−1019.97			−1.30598
$782$		0			0
$783$	−144.420	−	43.4228i	−0.184445	−	0.0554570i
$784$		0			0
$785$			613.278i			0.781246i
$786$		0			0
$787$	649.685			0.825520			0.412760	−	0.910840i	$-0.364565\pi$
$787$	649.685			0.825520			0.412760	+	0.910840i	$0.364565\pi$
$788$		0			0
$789$	702.230	+	68.5838i	0.890025	+	0.0869250i
$790$		0			0
$791$			633.358i			0.800706i
$792$		0			0
$793$	−1077.17			−1.35835
$794$		0			0
$795$	24.1491	−	247.262i	0.0303762	−	0.311022i
$796$		0			0
$797$		−	547.802i		−	0.687329i	−0.939092	−	0.343665i	$-0.888332\pi$
$797$		−	547.802i		−	0.687329i	0.939092	−	0.343665i	$-0.111668\pi$
$798$		0			0
$799$	−9.80522			−0.0122719
$800$		0			0
$801$	−302.463	+	1533.69i	−0.377607	+	1.91472i
$802$		0			0
$803$			562.338i			0.700296i
$804$		0			0
$805$	148.961			0.185044
$806$		0			0
$807$	660.059	+	64.4652i	0.817916	+	0.0798825i
$808$		0			0
$809$		−	719.685i		−	0.889598i	−0.895630	−	0.444799i	$-0.853275\pi$
$809$		−	719.685i		−	0.889598i	0.895630	−	0.444799i	$-0.146725\pi$
$810$		0			0
$811$	−775.519			−0.956251			−0.478125	−	0.878292i	$-0.658684\pi$
$811$	−775.519			−0.956251			−0.478125	+	0.878292i	$0.658684\pi$
$812$		0			0
$813$	78.3934	−	802.669i	0.0964248	−	0.987293i
$814$		0			0
$815$		−	582.085i		−	0.714214i
$816$		0			0
$817$	−1085.32			−1.32842
$818$		0			0
$819$	433.879	+	85.5666i	0.529767	+	0.104477i
$820$		0			0
$821$			1383.11i			1.68466i	0.538960	+	0.842331i	$0.318817\pi$
$821$			1383.11i			1.68466i	−0.538960	+	0.842331i	$0.681183\pi$
$822$		0			0
$823$	536.474			0.651852			0.325926	−	0.945395i	$-0.394324\pi$
$823$	536.474			0.651852			0.325926	+	0.945395i	$0.394324\pi$
$824$		0			0
$825$	266.774	+	26.0547i	0.323363	+	0.0315815i
$826$		0			0
$827$			1652.28i			1.99792i	0.0455686	+	0.998961i	$0.485490\pi$
$827$			1652.28i			1.99792i	−0.0455686	+	0.998961i	$0.514510\pi$
$828$		0			0
$829$	−999.416			−1.20557			−0.602784	−	0.797904i	$-0.705942\pi$
$829$	−999.416			−1.20557			−0.602784	+	0.797904i	$0.705942\pi$
$830$		0			0
$831$	42.1950	−	432.034i	0.0507761	−	0.519897i
$832$		0			0
$833$		−	23.0947i		−	0.0277247i
$834$		0			0
$835$	402.378			0.481890
$836$		0			0
$837$	−412.872	+	1373.17i	−0.493276	+	1.64059i
$838$		0			0
$839$		−	1394.04i		−	1.66155i	−0.556611	−	0.830773i	$-0.687899\pi$
$839$		−	1394.04i		−	1.66155i	0.556611	−	0.830773i	$-0.312101\pi$
$840$		0			0
$841$	809.803			0.962905
$842$		0			0
$843$	565.586	+	55.2384i	0.670920	+	0.0655260i
$844$		0			0
$845$			106.802i			0.126393i
$846$		0			0
$847$	−885.045			−1.04492
$848$		0			0
$849$	−122.657	+	1255.89i	−0.144473	+	1.47925i
$850$		0			0
$851$			772.602i			0.907875i
$852$		0			0
$853$	−1293.04			−1.51587			−0.757935	−	0.652330i	$-0.773792\pi$
$853$	−1293.04			−1.51587			−0.757935	+	0.652330i	$0.773792\pi$
$854$		0			0
$855$	−103.518	+	524.907i	−0.121074	+	0.613926i
$856$		0			0
$857$		−	1513.48i		−	1.76602i	−0.469357	−	0.883008i	$-0.655514\pi$
$857$		−	1513.48i		−	1.76602i	0.469357	−	0.883008i	$-0.344486\pi$
$858$		0			0
$859$	767.814			0.893846			0.446923	−	0.894572i	$-0.352520\pi$
$859$	767.814			0.893846			0.446923	+	0.894572i	$0.352520\pi$
$860$		0			0
$861$	903.724	+	88.2629i	1.04962	+	0.102512i
$862$		0			0
$863$		−	29.2279i		−	0.0338677i	−0.999857	−	0.0169339i	$-0.994610\pi$
$863$		−	29.2279i		−	0.0338677i	0.999857	−	0.0169339i	$-0.00539048\pi$
$864$		0			0
$865$	−248.795			−0.287625
$866$		0			0
$867$	84.0914	−	861.012i	0.0969913	−	0.993093i
$868$		0			0
$869$			38.6078i			0.0444279i
$870$		0			0
$871$	−33.3120			−0.0382457
$872$		0			0
$873$	809.157	+	159.576i	0.926870	+	0.182791i
$874$		0			0
$875$			49.8942i			0.0570220i
$876$		0			0
$877$	−1070.98			−1.22119			−0.610595	−	0.791943i	$-0.709070\pi$
$877$	−1070.98			−1.22119			−0.610595	+	0.791943i	$0.709070\pi$
$878$		0			0
$879$	−847.452	−	82.7671i	−0.964109	−	0.0941605i
$880$		0			0
$881$			489.535i			0.555659i	0.960630	+	0.277829i	$0.0896150\pi$
$881$			489.535i			0.555659i	−0.960630	+	0.277829i	$0.910385\pi$
$882$		0			0
$883$	−742.175			−0.840515			−0.420257	−	0.907405i	$-0.638060\pi$
$883$	−742.175			−0.840515			−0.420257	+	0.907405i	$0.638060\pi$
$884$		0			0
$885$	−39.8364	+	407.885i	−0.0450129	+	0.460887i
$886$		0			0
$887$		−	1441.82i		−	1.62550i	−0.582612	−	0.812751i	$-0.697969\pi$
$887$		−	1441.82i		−	1.62550i	0.582612	−	0.812751i	$-0.302031\pi$
$888$		0			0
$889$	503.924			0.566844
$890$		0			0
$891$	549.533	−	1339.06i	0.616759	−	1.50287i
$892$		0			0
$893$			328.282i			0.367617i
$894$		0			0
$895$	−174.072			−0.194494
$896$		0			0
$897$	−490.758	−	47.9303i	−0.547110	−	0.0534340i
$898$		0			0
$899$			296.629i			0.329954i
$900$		0			0
$901$	−29.4079			−0.0326392
$902$		0			0
$903$	−53.1272	+	543.969i	−0.0588341	+	0.602402i
$904$		0			0
$905$			534.090i			0.590154i
$906$		0			0
$907$	−352.397			−0.388531			−0.194265	−	0.980949i	$-0.562232\pi$
$907$	−352.397			−0.388531			−0.194265	+	0.980949i	$0.562232\pi$
$908$		0			0
$909$	202.614	−	1027.39i	0.222898	−	1.13024i
$910$		0			0
$911$		−	184.505i		−	0.202531i	−0.994859	−	0.101265i	$-0.967711\pi$
$911$		−	184.505i		−	0.202531i	0.994859	−	0.101265i	$-0.0322891\pi$
$912$		0			0
$913$	−233.574			−0.255831
$914$		0			0
$915$	−653.149	−	63.7903i	−0.713824	−	0.0697162i
$916$		0			0
$917$		−	478.545i		−	0.521859i
$918$		0			0
$919$	666.062			0.724769			0.362384	−	0.932029i	$-0.381963\pi$
$919$	666.062			0.724769			0.362384	+	0.932029i	$0.381963\pi$
$920$		0			0
$921$	48.9843	−	501.550i	0.0531860	−	0.544571i
$922$		0			0
$923$			628.479i			0.680909i
$924$		0			0
$925$	−258.782			−0.279765
$926$		0			0
$927$	1607.68	+	317.055i	1.73428	+	0.342022i
$928$		0			0
$929$		−	957.310i		−	1.03047i	−0.857048	−	0.515237i	$-0.827704\pi$
$929$		−	957.310i		−	1.03047i	0.857048	−	0.515237i	$-0.172296\pi$
$930$		0			0
$931$	−773.218			−0.830524
$932$		0			0
$933$	576.169	+	56.2720i	0.617544	+	0.0603130i
$934$		0			0
$935$		−	31.7285i		−	0.0339342i
$936$		0			0
$937$	1108.63			1.18317			0.591585	−	0.806243i	$-0.298502\pi$
$937$	1108.63			1.18317			0.591585	+	0.806243i	$0.298502\pi$
$938$		0			0
$939$	46.9655	−	480.879i	0.0500165	−	0.512119i
$940$		0			0
$941$		−	50.4523i		−	0.0536157i	−0.999641	−	0.0268078i	$-0.991466\pi$
$941$		−	50.4523i		−	0.0536157i	0.999641	−	0.0268078i	$-0.00853422\pi$
$942$		0			0
$943$	−1012.45			−1.07364
$944$		0			0
$945$	258.018	+	77.5783i	0.273035	+	0.0820935i
$946$		0			0
$947$		−	883.676i		−	0.933132i	−0.884486	−	0.466566i	$-0.845491\pi$
$947$		−	883.676i		−	0.933132i	0.884486	−	0.466566i	$-0.154509\pi$
$948$		0			0
$949$	346.497			0.365118
$950$		0			0
$951$	1536.97	+	150.109i	1.61616	+	0.157844i
$952$		0			0
$953$		−	1238.78i		−	1.29988i	−0.759987	−	0.649938i	$-0.774795\pi$
$953$		−	1238.78i		−	1.29988i	0.759987	−	0.649938i	$-0.225205\pi$
$954$		0			0
$955$	396.340			0.415015
$956$		0			0
$957$	29.1055	−	298.011i	0.0304132	−	0.311401i
$958$		0			0
$959$			302.317i			0.315242i
$960$		0			0
$961$	1859.39			1.93485
$962$		0			0
$963$	179.063	−	907.967i	0.185943	−	0.942852i
$964$		0			0
$965$		−	635.611i		−	0.658665i
$966$		0			0
$967$	1221.47			1.26315			0.631575	−	0.775314i	$-0.282409\pi$
$967$	1221.47			1.26315			0.631575	+	0.775314i	$0.282409\pi$
$968$		0			0
$969$	63.0304	+	6.15592i	0.0650469	+	0.00635286i
$970$		0			0
$971$		−	1142.08i		−	1.17619i	−0.808792	−	0.588095i	$-0.799878\pi$
$971$		−	1142.08i		−	1.17619i	0.808792	−	0.588095i	$-0.200122\pi$
$972$		0			0
$973$	−630.148			−0.647634
$974$		0			0
$975$	16.0542	−	164.379i	0.0164659	−	0.168594i
$976$		0			0
$977$			364.738i			0.373324i	0.982424	+	0.186662i	$0.0597669\pi$
$977$			364.738i			0.373324i	−0.982424	+	0.186662i	$0.940233\pi$
$978$		0			0
$979$	−3103.81			−3.17039
$980$		0			0
$981$	−666.004	−	131.344i	−0.678903	−	0.133888i
$982$		0			0
$983$			40.0459i			0.0407384i	0.999793	+	0.0203692i	$0.00648417\pi$
$983$			40.0459i			0.0407384i	−0.999793	+	0.0203692i	$0.993516\pi$
$984$		0			0
$985$	−546.364			−0.554684
$986$		0			0
$987$	164.536	+	16.0696i	0.166703	+	0.0162812i
$988$		0			0
$989$		−	609.410i		−	0.616188i
$990$		0			0
$991$	−910.307			−0.918574			−0.459287	−	0.888288i	$-0.651895\pi$
$991$	−910.307			−0.918574			−0.459287	+	0.888288i	$0.651895\pi$
$992$		0			0
$993$	18.5891	−	190.334i	0.0187202	−	0.191676i
$994$		0			0
$995$		−	165.496i		−	0.166328i
$996$		0			0
$997$	358.437			0.359516			0.179758	−	0.983711i	$-0.442469\pi$
$997$	358.437			0.359516			0.179758	+	0.983711i	$0.442469\pi$
$998$		0			0
$999$	−402.369	+	1338.24i	−0.402772	+	1.33958i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.3.l.a.41.5	✓	8
3.2	odd	2	inner	120.3.l.a.41.6	yes	8
4.3	odd	2		240.3.l.d.161.4		8
5.2	odd	4		600.3.c.d.449.1		16
5.3	odd	4		600.3.c.d.449.16		16
5.4	even	2		600.3.l.f.401.4		8
8.3	odd	2		960.3.l.g.641.5		8
8.5	even	2		960.3.l.h.641.4		8
12.11	even	2		240.3.l.d.161.3		8
15.2	even	4		600.3.c.d.449.15		16
15.8	even	4		600.3.c.d.449.2		16
15.14	odd	2		600.3.l.f.401.3		8
20.3	even	4		1200.3.c.m.449.1		16
20.7	even	4		1200.3.c.m.449.16		16
20.19	odd	2		1200.3.l.x.401.5		8
24.5	odd	2		960.3.l.h.641.3		8
24.11	even	2		960.3.l.g.641.6		8
60.23	odd	4		1200.3.c.m.449.15		16
60.47	odd	4		1200.3.c.m.449.2		16
60.59	even	2		1200.3.l.x.401.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.5	✓	8	1.1	even	1	trivial
120.3.l.a.41.6	yes	8	3.2	odd	2	inner
240.3.l.d.161.3		8	12.11	even	2
240.3.l.d.161.4		8	4.3	odd	2
600.3.c.d.449.1		16	5.2	odd	4
600.3.c.d.449.2		16	15.8	even	4
600.3.c.d.449.15		16	15.2	even	4
600.3.c.d.449.16		16	5.3	odd	4
600.3.l.f.401.3		8	15.14	odd	2
600.3.l.f.401.4		8	5.4	even	2
960.3.l.g.641.5		8	8.3	odd	2
960.3.l.g.641.6		8	24.11	even	2
960.3.l.h.641.3		8	24.5	odd	2
960.3.l.h.641.4		8	8.5	even	2
1200.3.c.m.449.1		16	20.3	even	4
1200.3.c.m.449.2		16	60.47	odd	4
1200.3.c.m.449.15		16	60.23	odd	4
1200.3.c.m.449.16		16	20.7	even	4
1200.3.l.x.401.5		8	20.19	odd	2
1200.3.l.x.401.6		8	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	120.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.291610 - 2.98579i) q^{3} -2.23607i q^{5} +4.46268 q^{7} +(-8.82993 - 1.74137i) q^{9} +O(q^{10})\)	\(q+(0.291610 - 2.98579i) q^{3} -2.23607i q^{5} +4.46268 q^{7} +(-8.82993 - 1.74137i) q^{9} -17.8696i q^{11} -11.0107 q^{13} +(-6.67644 - 0.652060i) q^{15} +0.794055i q^{17} +26.5852 q^{19} +(1.30136 - 13.3246i) q^{21} +14.9276i q^{23} -5.00000 q^{25} +(-7.77428 + 25.8565i) q^{27} +5.58545i q^{29} +53.1074 q^{31} +(-53.3549 - 5.21095i) q^{33} -9.97885i q^{35} +51.7565 q^{37} +(-3.21084 + 32.8758i) q^{39} +67.8236i q^{41} -40.8243 q^{43} +(-3.89383 + 19.7443i) q^{45} +12.3483i q^{47} -29.0845 q^{49} +(2.37088 + 0.231554i) q^{51} +37.0351i q^{53} -39.9576 q^{55} +(7.75251 - 79.3779i) q^{57} -61.0932i q^{59} +97.8289 q^{61} +(-39.4051 - 7.77119i) q^{63} +24.6208i q^{65} +3.02541 q^{67} +(44.5708 + 4.35305i) q^{69} -57.0787i q^{71} -31.4690 q^{73} +(-1.45805 + 14.9290i) q^{75} -79.7461i q^{77} -2.16053 q^{79} +(74.9352 + 30.7524i) q^{81} -13.0710i q^{83} +1.77556 q^{85} +(16.6770 + 1.62877i) q^{87} -173.692i q^{89} -49.1374 q^{91} +(15.4866 - 158.568i) q^{93} -59.4463i q^{95} -91.6381 q^{97} +(-31.1176 + 157.787i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9	\( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} - 8 q^{13} - 8 q^{19} + 28 q^{21} - 40 q^{25} + 20 q^{27} + 120 q^{31} - 112 q^{33} + 8 q^{37} - 72 q^{39} - 328 q^{43} - 60 q^{45} + 64 q^{49} + 64 q^{51} - 40 q^{55} + 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} + 32 q^{73} + 20 q^{75} + 88 q^{79} + 224 q^{81} - 152 q^{87} + 560 q^{91} - 368 q^{93} + 144 q^{97} + 32 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 + 16 * q^7 + 20 * q^9 - 8 * q^13 - 8 * q^19 + 28 * q^21 - 40 * q^25 + 20 * q^27 + 120 * q^31 - 112 * q^33 + 8 * q^37 - 72 * q^39 - 328 * q^43 - 60 * q^45 + 64 * q^49 + 64 * q^51 - 40 * q^55 + 72 * q^57 + 8 * q^61 + 88 * q^63 + 152 * q^67 + 100 * q^69 + 32 * q^73 + 20 * q^75 + 88 * q^79 + 224 * q^81 - 152 * q^87 + 560 * q^91 - 368 * q^93 + 144 * q^97 + 32 * q^99

Introduction

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